Building Krylov complexity from circuit complexity

Journal Article · 01 October 2024 · Physical Review Research

DOI: https://doi.org/10.1103/PhysRevResearch.6.L042001 · OSTI ID:2452844

Lv, Chenwei ; Zhang, Ren ; Zhou, Qi

Krylov complexity has emerged as a probe of operator growth in a wide range of nonequilibrium quantum dynamics. However, a fundamental issue remains in such studies: the definition of the distance between basis states in Krylov space is ambiguous. Here we show that Krylov complexity can be rigorously established from circuit complexity when dynamical symmetries exist. Whereas circuit complexity characterizes the geodesic distance in a multidimensional operator space, Krylov complexity measures the height of the final operator in a particular direction. The geometric representation of circuit complexity thus unambiguously designates the distance between basis states in Krylov space. This geometric approach also applies to time-dependent Liouvillian superoperators, where a single Krylov complexity is no longer sufficient. Multiple Krylov complexity may be exploited jointly to fully describe operator dynamics. Published by the American Physical Society 2024

• Sponsoring Organization: USDOE
• Grant/Contract Number: SC0019202
• OSTI ID: 2452844
• Journal Information: Physical Review Research, Journal Name: Physical Review Research Journal Issue: 4 Vol. 6; ISSN PPRHAI; ISSN 2643-1564
• Publisher: American Physical SocietyCopyright Statement
• Country of Publication: United States
• Language: English